Geometric decomposition of the d-dimensional hard-sphere partition function

Abstract

We introduce a geometric decomposition of the hard-sphere partition function. Using a close-packing-inspired geometric bound on the available insertion volume, made rigorous when a corresponding local density certificate is available, we establish a reference upper bound $Q^{\ast }$ on the configurational integral. Factoring this upper bound out of the statistical geometric partition function found by Speedy yields a new form for the $d$-dimensional partition function, $Q(N,V,T)=Q^{\ast }\exp (-N\mathcal{J})$, where $\mathcal{J}$ depends strictly on the boundary-to-volume ratio of the voids and the close-packing density. Overall, this work provides an exact reformulation of hard-sphere thermodynamics, thus deepening our statistical geometric understanding of hard spheres.

arXiv commentmain text: 6 pages, 3 figures plus supplementary information